We provide a formal treatment of security of digital signatures against *subversion attacks* (SAs).
Our model of subversion generalizes previous work in several directions, and is inspired by the proliferation of software attacks (e.g.,\\ malware and buffer overflow attacks), and by the recent revelations of Edward Snowden about intelligence agencies trying to surreptitiously sabotage cryptographic algorithms.
The main security requirement we put forward demands that a signature scheme should remain unforgeable even in the presence of an attacker applying SAs (within a certain class of allowed attacks) in a fully-adaptive and *continuous* fashion.
Previous notions---e.g.,\\ the notion of security against algorithm-substitution attacks introduced by Bellare et al. (CRYPTO \'14) for symmetric encryption---were non-adaptive and non-continuous.
In this vein, we show both positive and negative results for the goal of constructing subversion-resilient signature schemes.
-Negative results. As our main negative result, we show that a broad class of randomized signature schemes is unavoidably insecure against SAs, even if using just a single bit of randomness.
This improves upon earlier work that was only able to attack schemes with larger randomness space. When designing our new attack we consider undetectability as an explicit adversarial goal, meaning that the end-users (even the ones knowing the signing key) should not be able to detect that the signature scheme was subverted.
-Positive results. We complement the above negative results by showing that signature schemes with *unique* signatures are subversion-resilient against all attacks that meet a basic undetectability requirement. A similar result was shown by Bellare et al. for symmetric encryption, who proved the necessity to rely on *stateful* schemes; in contrast unique signatures are *stateless*, and in fact they are among the fastest and most established digital signatures available.
We finally show that it is possible to devise signature schemes secure against arbitrary tampering with the computation, by making use of an un-tamperable cryptographic reverse firewall (Mironov and Stephens-Davidowitz, EUROCRYPT \'15), i.e., an algorithm that \"sanitizes\" any signature given as input (using only public information). The firewall we design allows to successfully protect so-called re-randomizable signature schemes (which include unique signatures as special case).
As an additional contribution, we extend our model to consider multiple users and show implications and separations among the various notions we introduced.
While our study is mainly theoretical, due to its strong practical motivation, we believe that our results have important implications in practice and might influence the way digital signature schemes are selected or adopted in standards and protocols.